We consider a nonlinear Schroedinger equation (NLS) with random coefficients, in a regime of separation of scales corresponding to diffusion approximation. Our primary goal is to propose and study an efficient numerical scheme in this framework. We use a pseudo-spectral splitting scheme and we establish the order
of the global error. In particular we show that we can take an integration step larger than the smallest scale of the problem, here the correlation length of the random medium. Then, we study the asymptotic behavior of the numerical solution in the diffusion approximation regime.

Second order fully nonlinear PDEs arise from many areas in science
and engineering such as differential geometry, optimal control,
mass transportation, materials science, meteorology, geostrophic
fluid dynamics. They constitute the most difficult class of differential
equations to analyze analytically and to approximate numerically.
In the past two decades, enormous advances in the theoretical
analysis has been achieved, based on the viscosity solution theory,
for second order fully nonlinear PDEs. On the other hand, in contrast to
the success of the PDE analysis, numerical solutions for general
second order fully nonlinear PDEs is mostly an untouched area,
and computing viscosity solutions of second order fully nonlinear
PDEs has been impracticable.
In this talk, I shall first introduce a newly developed notion
of "moment solutions" and the "vanishing moment method" used
to construct such a solution for second order fully nonlinear PDEs,
and also discuss the convergence of the "vanishing moment method" and
the relationship between "moment solutions" and "viscosity solutions".
I shall then discuss how the "vanishing moment method" can be combined
with existing wealthy numerical methods/algorithms for 4th order
quasilinear PDEs to make it possible to construct practical
and convergent numerical methods for second order fully nonlinear PDEs.
Finally, I shall present some numerical experiment results for
the Monge-Ampere equation, the prescribed Gauss curvature equation,
the infinite-Laplace equation, and the nonlinear balance equation
(from geostrophic fluid dynamics) to demonstrate both convergence
and efficiency of the proposed numerical methodology. This is a joint
work with Michael Neilan of the University of Tennessee.

We develop general approaches that parameterize brain anatomical
surfaces with Riemann surface structure. All metric orientable surfaces
are Riemann surfaces and admit conformal structure. With harmonic
energy minimization, holomorphic 1-form and the Ricci flow methods, we
can parameterize brain surfaces with various canonical surfaces such as
sphere, Euclidean plane and punched hole disks. The resulting surface
subdivision and the parameterizations of the components are intrinsic
and stable. Our parameterization scheme offers a way to explicitly
match landmark curves in anatomical surfaces such as the cortex,
providing a surface-based framework to compare anatomy statistically and
to generate grids on surfaces for PDE-based signal processing. Various
applications of our research will also be discussed.